Which of the following condition is incorrect for matrix multiplication?

The correct option is AB = BA.

Let us analyze the rules and properties of matrix multiplication step-by-step to understand why this condition is incorrect in the general case.

1. Non-Commutativity of Matrix Multiplication (AB = BA is generally incorrect):
Unlike the multiplication of real numbers, matrix multiplication is not commutative. This means that if we have two matrices $A$ and $B$, the product $AB$ is not necessarily equal to $BA$.
In fact, even if $AB$ is defined (which requires the number of columns in $A$ to equal the number of rows in $B$), the product $BA$ might not even be defined. Even when both matrices are square and of the same size (so both $AB$ and $BA$ exist), they are generally not equal. Therefore, the statement $AB=BA$ is incorrect as a general rule for matrix multiplication.

2. Verifying the other options (which are correct properties):

• Associative Property: $A\left(BC\right)=\left(AB\right)C$
  Matrix multiplication is associative. As long as the dimensions of the matrices allow multiplication, grouping them differently does not change the result. Thus, this condition is correct.


• Distributive Property: $A(B+C)=AB+AC$
  Matrix multiplication distributes over matrix addition. Therefore, this condition is correct.


• Zero Product Property: $AB=0$ if either $A$ or $B$ is $0$ (Zero Matrix)
  If either matrix $A$ or matrix $B$ is a zero matrix (all its elements are zero), their product will always be a zero matrix. Note that the converse is not true (the product of two non-zero matrices can be zero), but the statement as written is correct.

Consequently, the incorrect condition for matrix multiplication is AB = BA.